The implied volatility smirk

Abstract

This paper provides an industry standard on how to quantify the shape of the implied volatility smirk in the equity index options market. Our local expansion method uses a second-order polynomial to describe the implied volatility–moneyness function and relates the coefficients of the polynomial to the properties of the implied risk-neutral distribution of the equity index return. We present a formal, two-way representation of the link between the level, slope and curvature of the implied volatility smirk and the risk-neutral standard deviation, skewness and excess kurtosis. We then propose a new semi-analytical method to calibrate option-pricing models based on the quantified implied volatility smirk, and investigate the applicability of two option-pricing models.

Keywords:

• Option pricing
• Implied volatility smirk
• Risk-neutral skewness and excess kurtosis
• Term structure

Acknowledgments

We are especially grateful to the two referees Mark Schroder and Liuren Wu whose helpful comments substantially improved the paper. We also acknowledge helpful comments from Jean-Philippe Bouchaud (co-editor-in-chief), Charles Cao, Peter Carr, Andrew P. Carverhill, Eric C. Chang, Bruno Dupire, Patrick S. Hagan, Jing-zhi Huang (our CICF discussant), Jun Pan, James J.-D. Wang, Chu Zhang, and seminar participants at the University of Hong Kong (HKU), Hong Kong University of Science and Technology (HKUST), Bloomberg, L. P., Claremont Graduate University, Shanghai University of Finance and Economics, Tongji University, University of Science and Technology of China, and 2005 China International Conference in Finance (CICF 2005) in Kunming. The research for this paper was partially supported by HKUST under the Direct Allocation Grant (Project No. DAG03/04.BM53), by HKU under the Small Project Funding scheme (Project No. 200507176196), and by the Research Grants Council of Hong Kong under the CERG grant (Project No. HKU 7427/06H).

Notes

†Both terms, implied volatility smirk and skew are used in the finance literature and the financial industry with slightly different meaning. A smirk can be regarded as the superposition of a skew and a smile.

‡The model is known as the SABR model, which stands for stochastic alpha beta rho (Hagan et al. 2002).

†In this paper, at the money is defined as the point that the strike price, K, is equal to the implied forward price, F ₀.

†These dates were randomly picked when we prepared the first draft of the paper.

†We follow the practice set up by the CBOE in computing the new volatility index, VIX. The definition of the VIX and the methodology of computing it from options prices are clearly described in the CBOE white paper, available at: http://www.cboe.com/micro/vix/vixwhite.pdf. The CBOE started trading VIX futures on 26 March 2004, and VIX options on 24 February 2006. Carr and Wu (2006), Zhang and Zhu (2006), and Zhu and Zhang (2007) study the price of VIX futures and their relationship with the underlying S&P 500 index options.

†The term structure of implied volatility has been studied by Heynen et al. (1994), Xu and Taylor (1994), Zhu and Avellaneda (1997), and Das and Sundaram (1999).

†For example, Navatte and Villa (2000) study the information content of implied volatility, skewness and kurtosis by using long-term CAC 40 options.

†Following the convention set up by CBOE in computing the new VIX index, with 8 days left to expiration, we roll to the second contract month in order to minimize pricing anomalies that might occur close to expiration.

†The skewness, , and excess kurtosis, , of a random number, x, are defined by where , i = 1,2,3,4 are the first four cumulants, given by , , , .

‡An explanation of the Edgeworth series expansion method can be found in Chapter 3 (p. 25) in Kolassa's (1997) book. A moment expansion approach to option pricing was studied by Airoldi (2005).

§Here we use Edgeworth series expansion to expand the unknown return distribution near a normal distribution. Jarrow and Rudd (1982) expand the unknown price distribution near a lognormal distribution and find a different option-pricing formula, see e.g. Corrado and Su (1997) for an empirical test of Jarrow and Rudd's (1982) model with SPX options. It seems to us that expanding the return distribution is more natural and consistent with the later advanced option-pricing models that model stock returns with a Lévy process or a time-changed Lévy process, see e.g. Carr and Wu (2004).

†The formula was presented by Backus et al. (1997) in the context of currency options.

‡The modified Bessel function of the first kind of order, ν, is a solution of the ordinary differential equation The function can be written in a series form as follows:

§The probability density function of a non-central chi-square distribution can be written in terms of the modified Bessel function as follows (Johnson and Kotz 1970):

¶The complementry non-central chi-square distribution function, Q(z;n,λ), satisfies the following identity .